Question

In a certain government office, there are 400 employee, there are 150 men, 276 University graduates, 212 married persons, 94 male University graduates, 151 married university graduates, 119 married men, 72 married male graduates. Find the number of single women who are not University graduates.

Answer 1

Let $U$ be set of all employees, $M$ - men, $A$ - university graduates, $B$ - married persons.

We know that $|U|=400, \ |M|=150, \ |A|=276, \ |B|=212$

$M \cap A$ - male university graduates, $|M\cap A| =94$

$A\cap B$ - married university graduates, $|A\cap B|=151$

$M\cap B$ - married men, $|M\cap B| =119$

$M\cap A\cap B$ - married male graduates, $| M\cap A\cap B|=72$

We need to find the number of single women who are not university graduates.

So, these persons aren’t men, aren’t married and aren’t university graduates.

Set of of single women who are not university graduates can be represented as $M ^\prime \cap A ^\prime \cap B ^\prime$ . Now we will find $|M ^\prime\cap A ^\prime\cap B ^\prime |$ .

$M ^\prime\cap A ^\prime \cap B ^\prime =(M\cup A\cup B) ^\prime$ (It is De Morgan's law)

$|M ^\prime\cap A ^\prime\cap B ^\prime|=|(M\cup A\cup B) ^\prime|=|U|-|M\cup A \cup B|$

$|M\cup A\cup B|=|M|+|A|+|B|-|M\cap A|-|A\cap B|-|M\cap B|+|M\cap A\cap B|$

( It is inclusion–exclusion principle )

$|M\cup A\cup B|=150+276+212-94-151-119+72=346$

$|U|-|M\cup A\cup B|=400-346=54$

Answer: there are 54 single women who are not University graduates.